Exponentially Smoothing the Skewed Laplace Distribution for Value‐at‐Risk Forecasting

Abstract

Value‐at‐risk (VaR) is a standard measure of market risk in financial markets. This paper proposes a novel, adaptive and efficient method to forecast both volatility and VaR. Extending existing exponential smoothing as well as GARCH formulations, the method is motivated from an asymmetric Laplace distribution, where skewness and heavy tails in return distributions, and their potentially time‐varying nature, are taken into account. The proposed volatility equation also involves novel time‐varying dynamics. Back‐testing results illustrate that the proposed method offers a viable, and more accurate, though conservative, improvement in forecasting VaR compared to a range of popular alternatives. Copyright © 2013 John Wiley & Sons, Ltd.

Score Driven Exponentially Weighted Moving Average and Value-at-Risk Forecasting

“…We also show that the SD-EWMA approach encompasses other proposals from the literature to model time-varying parameters, such as the normal based standard EWMA, the robust EWMA of Guermat and Harris [2002] based on the Laplace distribution, and the skewed EWMA of Gerlach et al [2013] based on the asymmetric Laplace distribution. Given that we are interested in modeling the time variation in financial risk measures, we explicitly develop an SD-EWMA model based on the fat-tailed skewed Student's t distribution; see for example Poon and Granger [2003] for stylized facts about financial returns.…”

“…The scheme also works for asymmetric distributions. For example, Gerlach et al [2013] introduces an EWMA scheme based on the asymmetric Laplace…”

“…Gerlach et al [2013] introduce EWMA type time variation in both σ t and p t , specified by the recursions…”

“…Again we notice from (15) that the original robust and asymmetric EWMA scheme of Gerlach et al [2013] can be interpreted as an SD-EWMA update if we set f t = σ t rather than f t = σ 2 t as in the original EWMA.…”

“…Compared to previous methods, such as Jensen andLunde [2001] andWilhelmsson [2009], the SD-EWMA approach has the distinct advantage that it provides a unifying framework that embeds previous proposals from the literature, such as Guermat and Harris [2002] and Gerlach et al [2013]. In addition, the generality of the SD-EWMA approach also allows for a straightforward generalization to higher dimensions, estimating score-driven versions of both volatilities, covariances and correlations, and other higherorder moments.…”